Sperner's Lemma - Multidimensional Case

Multidimensional Case

In the general case the lemma refers to a n-dimensional simplex

We consider a triangulation T which is a disjoint division of into smaller n-dimensional simplices. Denote the coloring function as f : S → {1,2,3,...,n,n+1}, where S is again the set of vertices of T. The rules of coloring are:

1. The vertices of the large simplex are colored with different colors, i. e. f(A_i) = i for 1 ≤ i ≤ n+1.
2. Vertices of T located on any k-dimensional subface

are colored only with the colors

Then there exists an odd number of simplices from T, whose vertices are colored with all n+1 colors. In particular, there must be at least one.